A New Blow-Up Criterion to a Singular Non-Newton Polytropic Filtration Equation

Abstract

In this paper, a singular non-Newton polytropic filtration equation under the initial-boundary value condition is revisited. The finite time blow-up results were discussed when the initial energy $E\left(u_0\right)$ was subcritical ($E\left(u_0\right)<d$), critical ($E\left(u_0\right)=d$), and supercritical ($E\left(u_0\right)>d$), with d being the potential depth by using the potential well method and some differential inequalities. The goal of this paper is to give a finite time blow-up result if $E\left(u_0\right)$ is independent of d. Moreover, the explicit upper bound of the blow-up time is obtained by the classical Levine’s concavity method, and the precise lower bound of the blow-up time is derived by applying an interpolation inequality.

1. Introduction

In this paper, we are concerned with the following initial-boundary value problem:

$$\left\{\begin{array}{cc}{\left|x\right|}^{-s}{u}_t-\mathrm{div}\left(\right|\nabla u^m{|}^{p-2}\nabla u^m)=u^q\hfill & \mathrm{in}\mathrm{\Omega }\times (0,T),\hfill \\ u(x,t)=0\hfill & \mathrm{on}\partial \mathrm{\Omega }\times (0,T),\hfill \\ u(x,0)=u_0\left(x\right)\hfill & \mathrm{for}x\in \mathrm{\Omega },\hfill \end{array}\right.$$

(1)

where the initial value $u_0\left(x\right)$ is a nonnegative and nontrivial function, $T\in (0,\infty ]$ is the maximal existence time of solutions, $\mathrm{\Omega }\subset {\mathbb{R}}^N(N>p)$ is a bounded domain with smooth boundary $\partial \mathrm{\Omega }$, $x=(x_1,x_2,\cdots ,x_N)\in {\mathbb{R}}^N$ with $\left|x\right|=\sqrt{x_1^2+x_1^2+\cdots +x_N^2}$, and the parameters satisfy

$$m\ge 1,p\ge 2,0\le s\le 1+\frac{1}{m},mp-m<q\le \frac{m(Np-N+p)}{N-p}.$$

(2)

Problem (1) in fact has its physical background. To be more precise, the volumetric moisture content $\theta \left(x\right)$, the macroscopic velocity $\overrightarrow{V}$, and the density of the fluid u, under the assumption that a compressible fluid flows in a homogeneous isotropic rigid porous medium, are governed by the following equation [1]:

$$\theta \left(x\right)u_t-\mathrm{div}\left(u\overrightarrow{V}\right)=f\left(u\right),$$

where $f\left(u\right)$ is the source. For the non-Newtonian fluid, provided that the fluid investigated is the polytropic gas, one obtains

$$\theta \left(x\right)u_t-c^{\alpha }\lambda \mathrm{div}\left(\right|\nabla u^m{|}^{p-2}\nabla u^m)=f\left(u\right),$$

(3)

where $c>0,m>0,\lambda >0,p\ge 2.$ Let $\theta \left(x\right)={\left|x\right|}^{-s}$ and $f\left(u\right)=u^q$ in (3); then, (1) can be deduced. For $m=1$ and $s=2$, in 2004, Tan in [2] considered the existence and asymptotic estimates of global solutions and the finite time blow-up of local solution based on the classical Hardy inequality [3]. Later on, Wang [4] extended the results obtained by Tan to $0\le s\le 2$, proved the existence of a global solution by the Hardy–Sobolev inequality [5], and found two sufficient conditions for blowing up in finite time by variational methods and classical concave methods. Zhou [6] discussed the global existence and finite time blow-up of solutions to problem (1) by the potential well method and the Hardy–Sobolev inequality when the initial energy is subcritical, i.e., $E\left(u_0\right)<d$. For $E\left(u_0\right)\ge d$, Xu and Zhou [7] discussed the behaviors of the solution by using the potential well method and some differential inequality techniques. Their results in fact extended previous one obtained by Hao and Zhou [8], where some blow-up conditions with $E\left(u_0\right)\ge d$ were obtained for $m=1$ and $p=2$ in problem (1).

The results above derived are not independent of the potential depth d. Naturally, we aim to present a new blow-up criterion when the initial energy is independent of d. In this paper, with the help of the Hardy–Sobolev inequality, we give a new blow-up result. Moreover, the upper and lower bounds of the blow-up time are derived. Our results extend the previous works in [9] and complement the results in [6,7].

2. Preliminaries

Throughout this paper, we denote by ${\parallel ·\parallel }_p$ and ${\parallel \nabla (·)\parallel }_p$ the norm on $L^p\left(\mathrm{\Omega }\right)$ and $W_0^{1,p}\left(\mathrm{\Omega }\right)$, respectively. Additionally, $(·,·)$ represents the inner product in $L^2\left(\mathrm{\Omega }\right)$. In order to present our main results, let us begin by introducing some definitions, notations, and lemmas obtained in [6,7].

It is well known that problem (1) is degenerate if $p>2$ at points where $\nabla u^m=0$, and therefore there is no classical solution in general. For this, we state the definition of the weak solution.

Definition 1. 

(Weak solution) A nonnegative function $u:=u(x,t)$ satisfying $u(x,0)=u_0\left(x\right)$ is called a weak solution of problem (1) on $\mathrm{\Omega }\times (0,T)$ if

$$u^m\in L^{\infty }(0,T;W_0^{1,p}\left(\mathrm{\Omega }\right)),u\in L^{m+q}(0,T;L^{m+q}\left(\mathrm{\Omega }\right)),{\int }_0^T\parallel {\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_t{\parallel }_2^{2}dt<+\infty ,$$

and u satisfies problem (1) in the distribution sense, that is

$${\left(\right|x|}^{-s}{u}_t,\varphi )+(|\nabla u^m{|}^{p-2}\nabla u^m{,\nabla \varphi )=(\left|u\right|}^q,\varphi )a.e.t\in (0,T)$$

for any $\varphi \in W_0^{1,p}\left(\mathrm{\Omega }\right)$.

Definition 2. 

(Finite time blow-up) Let u be a weak solution of problem (1) on $\mathrm{\Omega }\times (0,T)$. We say that u blows up at some finite time T if u exists for all $t\in [0,T)$ and

$$\underset{t\to T^{-}}{lim}{\int }_{\mathrm{\Omega }}{\left|x\right|}^{-s}{u}^{m+1}dx=+\infty .$$

Define $Q:=\left\{u:u^m\in W_0^{1,p}\left(\mathrm{\Omega }\right),u\in L^{m+q}\left(\mathrm{\Omega }\right)\right\}\backslash \left\{0\right\}$. Therefore, for any $u\in Q$, define the energy functional and Nehari functional by

$$E\left(u\right):=\frac{1}{mp}\parallel \nabla u^m{\parallel }_p^p-\frac{1}{m+q}{\parallel u^m\parallel }_{\frac{m+q}{m}}^{\frac{m+q}{m}},$$

(4)

$$H\left(u\right):=\parallel \nabla u^m{\parallel }_p^p-{\parallel u^m\parallel }_{\frac{m+q}{m}}^{\frac{m+q}{m}}.$$

(5)

Define the potential depth by

$$d:=\underset{u\in K}{inf}E\left(u\right)=\frac{m+q-mp}{mp(m+q)}{M}^{\frac{-p(m+q)}{m+q-mp}}>0,$$

where Nehari manifold

$$K:=\{u\in Q,H(u)=0\}\backslash \left\{0\right\},$$

and M is the optimal constant of the Sobolev embedding $W_0^{1,p}\left(\mathrm{\Omega }\right)\hookrightarrow L^{\frac{m+q}{m}}\left(\mathrm{\Omega }\right).$ In fact, M depends only on $\mathrm{\Omega },m,N,p$, and q such that for all $u\in Q$ it holds

$${\displaystyle \parallel }{u}^m{\parallel }_{\frac{m+q}{m}}\le M{\parallel \nabla u^m\parallel }_m.$$

Lemma 1. 

Let u be a weak solution of problem (1); then, the energy functional $E\left(u\right(t\left)\right)$ is non-increasing with respect to t. Moreover,

$$\frac{4}{{(m+1)}^2}{\int }_0^t\parallel {\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_{\tau }{\parallel }_2^{2}d\tau +E\left(u\left(t\right)\right)=E\left(u_0\right).$$

(6)

The following lemma is a descendant of Levine’s concavity method [10,11]. Further, the technique has also been revisited and presented in the book of Quittner and Souplet [12].

Lemma 2 

([10,11]). Suppose a positive, twice-differentiable function $\psi \left(t\right)$ satisfies the inequality

$${\psi }^{″}\left(t\right)\psi \left(t\right)-(1+\theta ){\left({\psi }^{\prime }\left(t\right)\right)}^2\ge 0,$$

where $\theta >0.$ If $\psi \left(0\right)>0,{\psi }^{\prime }\left(0\right)>0$, then $\psi \left(t\right)\to \infty$ as $t\to t_1\le t_2=\frac{\psi \left(0\right)}{\theta {\psi }^{\prime }\left(0\right)}$.

In order to prove our main results, we need the following Hardy–Sobolev inequality.

Lemma 3 

([5]). (Hardy–Sobolev inequality) Let ${\mathbb{R}}^N={\mathbb{R}}^k\times {\mathbb{R}}^{N-k}$, $2\le k\le N$ and $x=(y,z)\in {\mathbb{R}}^N={\mathbb{R}}^k\times {\mathbb{R}}^{N-k}$. For given $n,\beta$ satisfying $1<n<N$, $0\le \beta \le n$ and $\beta <k$, let $\gamma (\beta ,N,n)=n(N-\beta )/(N-n)$. Then, there exists a positive constant C depending on $\beta ,n,N$, and k such that for any $u\in W_0^{1,n}\left({\mathbb{R}}^N\right)$, it holds

$${\int }_{{\mathbb{R}}^N}\frac{{\left|u\left(x\right)\right|}^{\gamma }}{{\left|y\right|}^{\beta }}dx\le C{\left({\int }_{{\mathbb{R}}^N}{|\nabla u|}^{n}dx\right)}^{\frac{N-\beta }{N-n}}.$$

3. Main Results and Its Proof

As shown in [6,7], the finite time blow-up results were discussed when the initial energy $E\left(u_0\right)$ is subcritical ($E\left(u_0\right)<d$), critical ($E\left(u_0\right)=d$), and supercritical ($E\left(u_0\right)>d$), where d is the potential depth. Our first theorem will show a finite time blow-up result if $E\left(u_0\right)$ is independent of d and give a upper bound of the blow-up time.

Theorem 1. 

Let u be a weak solution of problem (1), and let (2) hold. If

$$0<C_{2}E\left(u_0\right)<\frac{L\left(0\right)}{m+1}-C_1$$

(7)

with $L\left(0\right)={\int }_{\mathrm{\Omega }}{\left|x\right|}^{-s}{\left|u_0\right|}^{m+1}dx$, $C_1=\frac{\tilde{C}}{m+1}$ and $C_2=\frac{mp(m+q)}{m+q-mp}\frac{\tilde{C}}{m+1}$, here

$$\tilde{C}=\left\{\begin{array}{cc}C\hfill & \mathit{if}\frac{N(m+1)}{mN+m+1-ms}=p,\hfill \\ \hfill {C\left|\mathrm{\Omega }\right|}^{\frac{mN+m+1-ms}{mN}-\frac{m+1}{mp}}& \hfill \mathit{if}\frac{N(m+1)}{mN+m+1-ms}<p.\end{array}\right.$$

then u blows up at some finite time T in the sense of Definition 1. Moreover, the upper of the blow-up time is given by

$$T\le \frac{8\tilde{C}L\left(0\right)}{F\left(0\right){(m+1)}^2{(q+m-2)}^2}\frac{mp(q+m-1)}{m+q-mp}$$

(8)

with $F\left(0\right)=\frac{L\left(0\right)}{m+1}-C_1-C_{2}E\left(u_0\right)>0.$

Proof. 

This proof follows some ideas in [9,13]. Firstly, we prove that u blows up in finite time. Suppose, on the contrary, that u is global, i.e., $T=+\infty$. For the sake of simplicity, define hereafter

$$L\left(t\right)={\int }_{\mathrm{\Omega }}{\left|x\right|}^{-s}{\left|u\right|}^{m+1}dx=\parallel {\left|x\right|}^{-\frac{s}{2}}{u}^{\frac{m+1}{2}}{\parallel }_2^2\mathrm{for}t\in [0,\infty ).$$

(9)

Then, for all $t\in [0,\infty )$, Hölder’s inequality and (6) imply

$$\begin{array}{cc}\hfill L^{\frac{1}{2}}\left(t\right)& =\parallel {\int }_0^t{\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_{\tau }d\tau +{\left|x\right|}^{-\frac{s}{2}}{u}_0^{\frac{m+1}{2}}{\parallel }_2\hfill \\ \hfill & \le {\int }_0^t\parallel {\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_{\tau }{\parallel }_{2}d\tau +L^{\frac{1}{2}}\left(0\right)\hfill \\ \hfill & \le t^{\frac{1}{2}}{\left({\int }_0^t\parallel {\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_{\tau }{\parallel }_2^{2}d\tau \right)}^{\frac{1}{2}}+L^{\frac{1}{2}}\left(0\right)\hfill \\ \hfill & \le t^{\frac{1}{2}}{\left[\frac{{(m+1)}^2}{4}\right]}^{\frac{1}{2}}{(E\left(u_0\right)-E\left(u\left(t\right)\right))}^{\frac{1}{2}}+L^{\frac{1}{2}}\left(0\right)\hfill \\ \hfill & \le t^{\frac{1}{2}}{\left[\frac{{(m+1)}^2}{4}\right]}^{\frac{1}{2}}{E}^{\frac{1}{2}}\left(u_0\right)+L^{\frac{1}{2}}\left(0\right),\hfill \end{array}$$

(10)

where we apply $0<E\left(u\left(t\right)\right)\le E\left(u_0\right)$ if u is a global solution. Here, we prove that $0<E\left(u\left(t\right)\right)\le E\left(u_0\right)$. Otherwise, there exists $t_0\in [0,\infty )$ such that $E\left(t_0\right)\le 0$. Then, by Remark 1.7 in [6], we know that u blows up in finite time, which is a contradiction. Multiplying the first equation of problem (1) with $u^m$ and integrating over $\mathrm{\Omega }$, and then recalling the definitions on $E\left(u\right)$ and $H\left(u\right)$ in (4) and (5), it follows that

$$\begin{array}{c}\hfill \frac{d}{dt}\frac{L\left(t\right)}{m+1}=-H\left(u\left(t\right)\right)=\frac{m+q-mp}{mp}{\parallel \nabla u^m\parallel }_p^p-(m+q)E\left(u\left(t\right)\right).\end{array}$$

(11)

On the other hand, applying the Hardy–Sobolev inequality in Lemma 3 yields

$$\frac{L\left(t\right)}{m+1}=\frac{1}{m+1}{\int }_{\mathrm{\Omega }}{\left|x\right|}^{-s}{\left(u^m\right)}^{\frac{m+1}{m}}dx\le \frac{C}{m+1}{\left({\int }_{\mathrm{\Omega }}{|\nabla u^m|}^{\frac{N(m+1)}{mN+m+1-ms}}dx\right)}^{\frac{mN+m+1-ms}{mN}}.$$

(12)

Noticing that $\frac{N(m+1)}{mN+m+1-ms}\le p$ and $p\ge 2\ge \frac{m+1}{m}$, it follows from the inequality $s\le s^{\alpha }+1$ with $s\ge 0,\alpha \ge 1$ that

$$\begin{array}{cc}\hfill & \frac{C}{m+1}{\left({\int }_{\mathrm{\Omega }}{|\nabla u^m|}^{\frac{N(m+1)}{mN+m+1-ms}}dx\right)}^{\frac{mN+m+1-ms}{mN}}\hfill \\ \hfill & \le \frac{\tilde{C}}{m+1}\parallel \nabla u^m{\parallel }_p^{\frac{m+1}{m}}\le \frac{\tilde{C}}{m+1}{\parallel \nabla u^m\parallel }_p^p+\frac{\tilde{C}}{m+1}\hfill \end{array}$$

(13)

with

$$\tilde{C}=\left\{\begin{array}{cc}C\hfill & \mathrm{if}\frac{N(m+1)}{mN+m+1-ms}=p,\hfill \\ {C\left|\mathrm{\Omega }\right|}^{\frac{mN+m+1-ms}{mN}-\frac{m+1}{mp}}\hfill & \mathrm{if}\frac{N(m+1)}{mN+m+1-ms}<p.\hfill \end{array}\right.$$

Therefore, let us combine (11) with (12) and (17) to obtain

$$\begin{array}{c}\hfill \frac{d}{dt}\frac{L\left(t\right)}{m+1}=-H\left(u\left(t\right)\right)\ge \frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}\left[\frac{L\left(t\right)}{m+1}-C_1-C_{2}E\left(u\left(t\right)\right)\right],\end{array}$$

(14)

here $C_1=\frac{\tilde{C}}{m+1}$ and $C_2=\frac{mp(m+q)}{m+q-mp}\frac{\tilde{C}}{m+1}$. Set

$$F\left(t\right)=\frac{L\left(t\right)}{m+1}-C_1-C_{2}E\left(u\left(t\right)\right),$$

(15)

then by Lemma 1 and (14)

$$\frac{d}{dt}F\left(t\right)\ge \frac{d}{dt}\frac{L\left(t\right)}{m+1}\ge \frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}F\left(t\right).$$

Since $F\left(0\right)=\frac{L\left(0\right)}{m+1}-C_1-C_{2}E\left(u_0\right)>0$, we obtain

$$F\left(t\right)\ge F\left(0\right)e^{\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}t}>0,$$

(16)

Therefore, by (15), one has

$$L\left(t\right)\ge (m+1)F\left(t\right)\ge (m+1)F\left(0\right)e^{\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}t},$$

which contradicts (10) for sufficiently large t. Thus, u blows up in finite time. Moreover, (14) and (16) imply that $L\left(t\right)$ is strictly increasing for $t\in [0,\infty )$.

Secondly, let us estimate the upper bound of T. For any $T^{*}\in (0,T)$, $\gamma >0$, and $\sigma >0$, define an auxiliary function

$$M\left(t\right)={\int }_0^t\frac{L\left(\tau \right)}{m+1}d\tau +(T-t)\frac{L\left(0\right)}{m+1}+\frac{\gamma }{2}{(t+\sigma )}^2\mathrm{for}t\in [0,T^{*}].$$

By a direct computation, one has

$$\begin{array}{c}\hfill M^{\prime }\left(t\right)=\frac{L\left(t\right)-L\left(0\right)}{m+1}+\gamma (t+\sigma )={\int }_0^t{\int }_{\mathrm{\Omega }}{\left|x\right|}^{-s}{\left|u\right|}^{m-1}u{u}_{\tau }dxd\tau +\gamma (t+\sigma )\mathrm{for}t\in [0,T^{*}],\end{array}$$

Further, recall (14) and (6); then,

$$\begin{array}{cc}\hfill M^{″}\left(t\right)& =\frac{d}{dt}\frac{L\left(t\right)}{m+1}+\gamma \ge \frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}\left[\frac{L\left(t\right)}{m+1}-C_1-C_{2}E\left(u\left(t\right)\right)\right]+\gamma \hfill \\ \hfill & =\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}[\frac{L\left(t\right)}{m+1}-C_1-C_{2}E\left(u_0\right)\hfill \\ \hfill & +C_2\frac{4}{{(m+1)}^2}{\int }_0^t\parallel {\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_{\tau }{\parallel }_2^{2}d\tau ]+\gamma \hfill \\ \hfill & \ge \frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}\left[F\left(0\right)+C_2\frac{4}{{(m+1)}^2}{\int }_0^t\parallel {\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_{\tau }{\parallel }_2^{2}d\tau \right]+\gamma \hfill \end{array}$$

for $t\in [0,T^{*}]$. Here, we have assumed that $L\left(t\right)$ is strictly increasing for $t\in [0,\infty )$. Applying the Cauchy–Schwarz inequality and Young’s inequality, one has

$$\begin{array}{cc}\hfill \xi \left(t\right):& =\left[{\int }_0^{t}L\left(\tau \right)d\tau +\gamma {(t+\sigma )}^2\right]\left[\frac{4}{{(m+1)}^2}{\int }_0^t\parallel {\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_{\tau }{\parallel }_2^{2}d\tau +\gamma \right]\hfill \\ \hfill & -{\left[{\int }_0^t{\int }_{\mathrm{\Omega }}{\left|x\right|}^{-s}{\left|u\right|}^{m-1}u{u}_{\tau }dxd\tau +\gamma (t+\sigma )\right]}^2\ge 0\mathrm{for}t\in [0,T^{*}].\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & M\left(t\right)M^{″}\left(t\right)-\frac{q+m}{2}{\left(M^{\prime }\left(t\right)\right)}^2\hfill \\ \hfill & \ge M\left(t\right)M^{″}\left(t\right)+\frac{q+m}{2}[\xi \left(t\right)-\left(2M\left(t\right)-\frac{2(T-t)L\left(0\right)}{m+1}\right)\hfill \\ \hfill & \times \left(\frac{4}{{(m+1)}^2}{\int }_0^t\parallel {\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_{\tau }{\parallel }_2^{2}d\tau +\gamma \right)]\hfill \\ \hfill & \ge M\left(t\right)\left[M^{″}\left(t\right)-(q+m)\left(\frac{4}{{(m+1)}^2}{\int }_0^t\parallel {\left|x\right|}^{-\frac{s}{2}}{\left(u^{\frac{m+1}{2}}\right)}_{\tau }{\parallel }_2^{2}d\tau +\gamma \right)\right]\hfill \\ \hfill & \ge M\left(t\right)\left[\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}F\left(0\right)-(q+m-1)\gamma \right]\ge 0\hfill \end{array}$$

(17)

for $t\in [0,T^{*}]$ and $\gamma \in \left(0,\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}\frac{1}{q+m-1}F\left(0\right)\right]$. It follows from Lemma 2 that

$$T^{*}\le \frac{2M\left(0\right)}{(q+m-2)M^{\prime }\left(0\right)}=\frac{2TL\left(0\right)}{(m+1)(q+m-2)\gamma \sigma }+\frac{\sigma }{q+m-2}.$$

Since the arbitrariness of $T^{*}<T$, for any $\gamma \in \left(0,\frac{mp-(m+q)}{mp}\frac{m+1}{\tilde{C}}\frac{1}{q+m-1}F\left(0\right)\right]$ and $\sigma >0$, one has

$$T\le \frac{2TL\left(0\right)}{(m+1)(q+m-2)\gamma \sigma }+\frac{\sigma }{q+m-2}.$$

(18)

Fix now ${\gamma }_0\in \left(0,\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}\frac{1}{q+m-1}F\left(0\right)\right]$; then, for any $\sigma \in \left(\frac{2L\left(0\right)}{(m+1)(q+m-2){\gamma }_0},+\infty \right)$, $0<\frac{2L\left(0\right)}{(m+1)(q+m-2){\gamma }_0}<1$ holds, which implies together with (18)

$$T\le \frac{(m+1){\gamma }_0{\sigma }^2}{(m+1)(q+m-2){\gamma }_0\sigma -2L\left(0\right)}.$$

Minimizing the right-hand side of the inequality above for $\sigma \in \left(\frac{2L\left(0\right)}{(m+1)(q+m-2){\gamma }_0},+\infty \right)$, one obtains

$$T\le \frac{8L\left(0\right)}{(m+1){(q+m-2)}^2{\gamma }_0}\mathrm{for}{\gamma }_0\in \left(0,\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}\frac{1}{q+m-1}F\left(0\right)\right].$$

(19)

Minimizing the right hand side of the inequality above for ${\gamma }_0\in \left(0,\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}\frac{1}{q+m-1}F\left(0\right)\right]$, we obtain (8). □

Next, we shall derive a lower bound for the blow-up time T by combining the interpolation inequality with the first order differential inequalities.

Theorem 2. 

Suppose that all conditions of Theorem 1 are fulfilled and $m+q-mp<(m+1)p/N$. Then, the lower bound of the blow-up time can be estimated by

$$T\ge \frac{L^{1-\kappa }\left(0\right)}{(m+1){\left[C_{*}^{\frac{\theta (m+q)}{m}}{\left[\mathit{diam}\left(\mathrm{\Omega }\right)\right]}^{\frac{s(1-\theta )(m+q)}{m+1}}\right]}^{\frac{mp}{mp-\theta (m+q)}}(\kappa -1)}$$

(20)

with $\theta =\left(\frac{m}{m+1}-\frac{m}{m+q}\right){\left(\frac{m}{m+1}-\frac{N-p}{Np}\right)}^{-1}\in (0,1)$, $\kappa =\left[\frac{(1-\theta )(m+q)}{m+1}\right]/[1-\frac{\theta (m+q)}{mp}]$, and $C_{*}$ is the optimal constant of embedding $W_0^{1,p}\left(\mathrm{\Omega }\right)\hookrightarrow L^{\frac{Np}{N-p}}\left(\mathrm{\Omega }\right)$.

Proof. 

By (14), one has

$$H\left(u_0\right)\le -\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}\left[\frac{L\left(0\right)}{m+1}-C_1-C_{2}E\left(u_0\right)\right]=-\frac{m+q-mp}{mp}\frac{m+1}{\tilde{C}}F\left(0\right)<0.$$

In what follows, we claim $H\left(u\right(t\left)\right)<0$ for $t\in [0,T)$. If not, $t_0\in (0,T)$ would exist such that $H\left(u\right(t\left)\right)<0$ for all $t\in [0,t_0)$ and $H\left(u\left(t_0\right)\right)=0$. Recalling that $L\left(t\right)$ is strictly increasing and $F\left(0\right)>0$, one has

$$C_1+C_{2}E\left(u_0\right)<\frac{L\left(0\right)}{m+1}<\frac{L\left(t_0\right)}{m+1}.$$

(21)

On the other hand, (6) and (14) with $t=t_0$ implies

$$C_1+C_{2}E\left(u_0\right)\ge C_1+C_{2}E\left(u\left(t_0\right)\right)\ge \frac{L\left(t_0\right)}{m+1},$$

which contradicts (21). Therefore, we illustrate that $H\left(u\right(t\left)\right)<0$ for $t\in [0,T)$.

Applying interpolation inequality, $W_0^{1,p}\left(\mathrm{\Omega }\right)\hookrightarrow L^{\frac{Np}{N-p}}\left(\mathrm{\Omega }\right)$ and $H\left(u\right(t\left)\right)<0$ for $t\in [0,T)$, it follows that

$$\begin{array}{cc}\hfill \parallel u^m{\parallel }_{\frac{m+q}{m}}^{\frac{m+q}{m}}& \le \parallel u^m{\parallel }_{\frac{Np}{N-p}}^{\frac{\theta (m+q)}{m}}\parallel u^m{\parallel }_{\frac{m+1}{m}}^{\frac{(1-\theta )(m+q)}{m}}\le C_{*}^{\frac{\theta (m+q)}{m}}\parallel \nabla u^m{\parallel }_p^{\frac{\theta (m+q)}{m}}{\parallel u^m\parallel }_{\frac{m+1}{m}}^{\frac{(1-\theta )(m+q)}{m}}\hfill \\ \hfill & <C_{*}^{\frac{\theta (m+q)}{m}}{\left(\parallel u^m{\parallel }_{\frac{m+q}{m}}^{\frac{m+q}{m}}\right)}^{\frac{\theta (m+q)}{mp}}{\left(\parallel u^m{\parallel }_{\frac{m+1}{m}}^{\frac{m+1}{m}}\right)}^{\frac{(1-\theta )(m+q)}{m+1}}\hfill \\ \hfill & =C_{*}^{\frac{\theta (m+q)}{m}}{\left[\mathrm{diam}\left(\mathrm{\Omega }\right)\right]}^{\frac{s(1-\theta )(m+q)}{m+1}}{\left(\parallel u^m{\parallel }_{\frac{m+q}{m}}^{\frac{m+q}{m}}\right)}^{\frac{\theta (m+q)}{mp}}{\left(L\left(t\right)\right)}^{\frac{(1-\theta )(m+q)}{m+1}}.\hfill \end{array}$$

(22)

Here, we have used the definition of $L\left(t\right)$ in (9). By recalling the value of $\theta$ and $m+q-mp<(m+1)p/N$, one has $1-\theta (m+q)/\left(mp\right)>0$, and $\kappa =\left[\frac{(1-\theta )(m+q)}{m+1}\right]/[1-\frac{\theta (m+q)}{mp}]>1.$ Therefore, it follows from (14) and (22) that

$$\begin{array}{cc}\hfill \frac{d}{dt}L\left(t\right)& =-(m+1)H\left(u\left(t\right)\right)\le m+1\parallel u^m{\parallel }_{\frac{m+q}{m}}^{\frac{m+q}{m}}\hfill \\ \hfill & <(m+1){\left[C_{*}^{\frac{\theta (m+q)}{m}}{\left[\mathrm{diam}\left(\mathrm{\Omega }\right)\right]}^{\frac{s(1-\theta )(m+q)}{m+1}}\right]}^{\frac{mp}{mp-\theta (m+q)}}{L}^{\kappa }\left(t\right)\hfill \end{array}$$

(23)

and $L\left(t\right)>0$ due to $H\left(u\right(t\left)\right)<0$ for $t\in [0,T)$. Further, (23) yields

$$\frac{1}{1-\kappa }(L^{1-\kappa }\left(t\right)-L^{1-\kappa }\left(0\right))\le (m+1){\left[C_{*}^{\frac{\theta (m+q)}{m}}{\left[\mathrm{diam}\left(\mathrm{\Omega }\right)\right]}^{\frac{s(1-\theta )(m+q)}{m+1}}\right]}^{\frac{mp}{mp-\theta (m+q)}}t.$$

By Theorem 1, we obtain $\underset{t\to T^{-}}{lim}L\left(t\right)=+\infty .$ Thus, (20) can be derived by letting $t\to T$. □

4. Conclusions

In this paper, we establish new results on the blow-up in finite time of weak solutions to problem (1). Previous blow-up results were obtained by the potential well method and some differential inequalities, where the potential depth d plays an important role. We give a finite time blow-up result if $E\left(u_0\right)$ is independent of d and obtain the upper bound and lower bound of the blow-up time by the classical Levine’s concavity method and an interpolation inequality. In fact, our results illustrate that the blow-up phenomenon will happen when the initial energy is arbitrarily high. Blow-up rate estimates are also of importance; therefore, considering blow-up rate estimates will be the focus of our next work.